Stability for Nash Equilibrium Problems

TL;DR

This paper analyzes the stability properties of the KKT solution mapping in Nash equilibrium problems, providing characterizations and conditions for regularity, localization, and robustness under perturbations.

Contribution

It offers new characterizations of strong regularity, localization, and stability of the KKT solution mapping in NEPs, including for quadratic programming cases.

Findings

Exact characterization of strong regularity of $S_{\rm KKT}$

Conditions for continuous differentiability of $S_{\rm KKT}$

Robustness of $E(p)$ and $S_{\rm KKT}$ under convex assumptions

Abstract

This paper is devoted to studying the stability properties of the Karush-Kuhn-Tucker (KKT) solution mapping $S_{\rm KKT}$ for Nash equilibrium problems (NEPs) with canonical perturbations. Firstly, we obtain an exact characterization of the strong regularity of $S_{\rm KKT}$ and a sufficient condition that is easy to verify. Secondly, we propose equivalent conditions for the continuously differentiable single-valued localization of $S_{\rm KKT}$. Thirdly, the isolated calmness of $S_{\rm KKT}$ is studied based on two conditions: Property A and Property B, and Property B proves to be sufficient for the robustness of both $E(p)$ and $S_{\rm KKT}$ under the convex assumptions, where $E(p)$ denotes the Nash equilibria at perturbation $p$. Furthermore, we establish that studying the stability properties of the NEP with canonical perturbations is equivalent to studying those of the NEP with only tilt perturbations based on the prior discussions. Finally, we provide detailed characterizations of stability for NEPs whose each individual player solves a quadratic programming (QP) problem.